(* Title: HOL/BNF/Tools/bnf_def_tactics.ML
Author: Dmitriy Traytel, TU Muenchen
Author: Jasmin Blanchette, TU Muenchen
Copyright 2012
Tactics for definition of bounded natural functors.
*)
signature BNF_DEF_TACTICS =
sig
val mk_collect_set_map_tac: thm list -> tactic
val mk_map_id: thm -> thm
val mk_map_comp: thm -> thm
val mk_map_cong_tac: Proof.context -> thm -> tactic
val mk_in_mono_tac: int -> tactic
val mk_map_wppull_tac: thm -> thm -> thm -> thm -> thm list -> tactic
val mk_set_map: thm -> thm
val mk_rel_Grp_tac: thm list -> thm -> thm -> thm -> thm -> thm list ->
{prems: thm list, context: Proof.context} -> tactic
val mk_rel_eq_tac: int -> thm -> thm -> thm -> tactic
val mk_rel_OO_tac: thm list -> thm -> thm -> thm -> thm -> thm list ->
{prems: thm list, context: Proof.context} -> tactic
val mk_rel_conversep_tac: thm -> thm -> tactic
val mk_rel_conversep_le_tac: thm list -> thm -> thm -> thm -> thm list ->
{prems: thm list, context: Proof.context} -> tactic
val mk_rel_mono_tac: thm list -> thm -> tactic
val mk_rel_mono_strong_tac: thm -> thm list -> {prems: 'a, context: Proof.context} -> tactic
val mk_map_transfer_tac: thm -> thm -> thm list -> thm -> thm ->
{prems: thm list, context: Proof.context} -> tactic
val mk_in_bd_tac: int -> thm -> thm -> thm -> thm -> thm list -> thm list -> thm -> thm -> thm ->
thm -> {prems: thm list, context: Proof.context} -> tactic
val mk_trivial_wit_tac: thm list -> thm list -> {prems: thm list, context: Proof.context} ->
tactic
end;
structure BNF_Def_Tactics : BNF_DEF_TACTICS =
struct
open BNF_Util
open BNF_Tactics
val ord_eq_le_trans = @{thm ord_eq_le_trans};
val ord_le_eq_trans = @{thm ord_le_eq_trans};
val conversep_shift = @{thm conversep_le_swap} RS iffD1;
fun mk_map_id id = mk_trans (fun_cong OF [id]) @{thm id_apply};
fun mk_map_comp comp = @{thm o_eq_dest_lhs} OF [mk_sym comp];
fun mk_map_cong_tac ctxt cong0 =
(hyp_subst_tac ctxt THEN' rtac cong0 THEN'
REPEAT_DETERM o (dtac meta_spec THEN' etac meta_mp THEN' atac)) 1;
fun mk_set_map set_map0 = set_map0 RS @{thm comp_eq_dest};
fun mk_in_mono_tac n = if n = 0 then rtac subset_UNIV 1
else (rtac subsetI THEN'
rtac CollectI) 1 THEN
REPEAT_DETERM (eresolve_tac [CollectE, conjE] 1) THEN
REPEAT_DETERM_N (n - 1)
((rtac conjI THEN' etac subset_trans THEN' atac) 1) THEN
(etac subset_trans THEN' atac) 1;
fun mk_collect_set_map_tac set_map0s =
(rtac (@{thm collect_o} RS trans) THEN' rtac @{thm arg_cong[of _ _ collect]} THEN'
EVERY' (map (fn set_map0 =>
rtac (mk_trans @{thm image_insert} @{thm arg_cong2[of _ _ _ _ insert]}) THEN'
rtac set_map0) set_map0s) THEN'
rtac @{thm image_empty}) 1;
fun mk_map_wppull_tac map_id0 map_cong0 map_wpull map_comp set_maps =
if null set_maps then
EVERY' [rtac @{thm wppull_id}, rtac map_wpull, rtac map_id0, rtac map_id0] 1
else EVERY' [REPEAT_DETERM o etac conjE, REPEAT_DETERM o dtac @{thm wppull_thePull},
REPEAT_DETERM o etac exE, rtac @{thm wpull_wppull}, rtac map_wpull,
REPEAT_DETERM o rtac @{thm wpull_thePull}, rtac ballI,
REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac conjI, rtac CollectI,
CONJ_WRAP' (fn thm => EVERY' [rtac (thm RS ord_eq_le_trans),
rtac @{thm image_subsetI}, rtac conjunct1, etac bspec, etac set_mp, atac])
set_maps,
CONJ_WRAP' (fn thm => EVERY' [rtac (map_comp RS trans), rtac (map_comp RS trans),
rtac (map_comp RS trans RS sym), rtac map_cong0,
REPEAT_DETERM_N (length set_maps) o EVERY' [rtac (o_apply RS trans),
rtac (o_apply RS trans RS sym), rtac (o_apply RS trans), rtac thm,
rtac conjunct2, etac bspec, etac set_mp, atac]]) [conjunct1, conjunct2]] 1;
fun mk_rel_Grp_tac rel_OO_Grps map_id0 map_cong0 map_id map_comp set_maps
{context = ctxt, prems = _} =
let
val n = length set_maps;
val rel_OO_Grps_tac = if null rel_OO_Grps then K all_tac else rtac (hd rel_OO_Grps RS trans);
in
if null set_maps then
unfold_thms_tac ctxt ((map_id0 RS @{thm Grp_UNIV_id}) :: rel_OO_Grps) THEN
rtac @{thm Grp_UNIV_idI[OF refl]} 1
else
EVERY' [rel_OO_Grps_tac, rtac @{thm antisym}, rtac @{thm predicate2I},
REPEAT_DETERM o
eresolve_tac [CollectE, exE, conjE, @{thm GrpE}, @{thm relcomppE}, @{thm conversepE}],
hyp_subst_tac ctxt, rtac @{thm GrpI}, rtac trans, rtac map_comp, rtac map_cong0,
REPEAT_DETERM_N n o EVERY' [rtac @{thm Collect_split_Grp_eqD}, etac @{thm set_mp}, atac],
rtac CollectI,
CONJ_WRAP' (fn thm => EVERY' [rtac (thm RS ord_eq_le_trans),
rtac @{thm image_subsetI}, rtac @{thm Collect_split_Grp_inD}, etac @{thm set_mp}, atac])
set_maps,
rtac @{thm predicate2I}, REPEAT_DETERM o eresolve_tac [@{thm GrpE}, exE, conjE],
hyp_subst_tac ctxt,
rtac @{thm relcomppI}, rtac @{thm conversepI},
EVERY' (map2 (fn convol => fn map_id0 =>
EVERY' [rtac @{thm GrpI}, rtac (box_equals OF [map_cong0, map_comp RS sym, map_id0]),
REPEAT_DETERM_N n o rtac (convol RS fun_cong),
REPEAT_DETERM o eresolve_tac [CollectE, conjE],
rtac CollectI,
CONJ_WRAP' (fn thm =>
EVERY' [rtac ord_eq_le_trans, rtac thm, rtac @{thm image_subsetI},
rtac @{thm convol_mem_GrpI}, etac set_mp, atac])
set_maps])
@{thms fst_convol snd_convol} [map_id, refl])] 1
end;
fun mk_rel_eq_tac n rel_Grp rel_cong map_id0 =
(EVERY' (rtac (rel_cong RS trans) :: replicate n (rtac @{thm eq_alt})) THEN'
rtac (rel_Grp RSN (2, @{thm box_equals[OF _ sym sym[OF eq_alt]]})) THEN'
(if n = 0 then rtac refl
else EVERY' [rtac @{thm arg_cong2[of _ _ _ _ "Grp"]},
rtac @{thm equalityI}, rtac subset_UNIV, rtac subsetI, rtac CollectI,
CONJ_WRAP' (K (rtac subset_UNIV)) (1 upto n), rtac map_id0])) 1;
fun mk_rel_mono_tac rel_OO_Grps in_mono =
let
val rel_OO_Grps_tac = if null rel_OO_Grps then K all_tac
else rtac (hd rel_OO_Grps RS ord_eq_le_trans) THEN'
rtac (hd rel_OO_Grps RS sym RSN (2, ord_le_eq_trans));
in
EVERY' [rel_OO_Grps_tac, rtac @{thm relcompp_mono}, rtac @{thm iffD2[OF conversep_mono]},
rtac @{thm Grp_mono}, rtac in_mono, REPEAT_DETERM o etac @{thm Collect_split_mono},
rtac @{thm Grp_mono}, rtac in_mono, REPEAT_DETERM o etac @{thm Collect_split_mono}] 1
end;
fun mk_rel_conversep_le_tac rel_OO_Grps rel_eq map_cong0 map_comp set_maps
{context = ctxt, prems = _} =
let
val n = length set_maps;
val rel_OO_Grps_tac = if null rel_OO_Grps then K all_tac
else rtac (hd rel_OO_Grps RS ord_eq_le_trans) THEN'
rtac (hd rel_OO_Grps RS sym RS @{thm arg_cong[of _ _ conversep]} RSN (2, ord_le_eq_trans));
in
if null set_maps then rtac (rel_eq RS @{thm leq_conversepI}) 1
else
EVERY' [rel_OO_Grps_tac, rtac @{thm predicate2I},
REPEAT_DETERM o
eresolve_tac [CollectE, exE, conjE, @{thm GrpE}, @{thm relcomppE}, @{thm conversepE}],
hyp_subst_tac ctxt, rtac @{thm conversepI}, rtac @{thm relcomppI}, rtac @{thm conversepI},
EVERY' (map (fn thm => EVERY' [rtac @{thm GrpI}, rtac sym, rtac trans,
rtac map_cong0, REPEAT_DETERM_N n o rtac thm,
rtac (map_comp RS sym), rtac CollectI,
CONJ_WRAP' (fn thm => EVERY' [rtac (thm RS ord_eq_le_trans),
etac @{thm flip_pred}]) set_maps]) [@{thm snd_fst_flip}, @{thm fst_snd_flip}])] 1
end;
fun mk_rel_conversep_tac le_conversep rel_mono =
EVERY' [rtac @{thm antisym}, rtac le_conversep, rtac @{thm xt1(6)}, rtac conversep_shift,
rtac le_conversep, rtac @{thm iffD2[OF conversep_mono]}, rtac rel_mono,
REPEAT_DETERM o rtac @{thm eq_refl[OF sym[OF conversep_conversep]]}] 1;
fun mk_rel_OO_tac rel_OO_Grps rel_eq map_cong0 map_wppull map_comp set_maps
{context = ctxt, prems = _} =
let
val n = length set_maps;
fun in_tac nthO_in = rtac CollectI THEN'
CONJ_WRAP' (fn thm => EVERY' [rtac (thm RS ord_eq_le_trans),
rtac @{thm image_subsetI}, rtac nthO_in, etac set_mp, atac]) set_maps;
val rel_OO_Grps_tac = if null rel_OO_Grps then K all_tac
else rtac (hd rel_OO_Grps RS trans) THEN'
rtac (@{thm arg_cong2[of _ _ _ _ "op OO"]} OF (replicate 2 (hd rel_OO_Grps RS sym)) RSN
(2, trans));
in
if null set_maps then rtac (rel_eq RS @{thm eq_OOI}) 1
else
EVERY' [rel_OO_Grps_tac, rtac @{thm antisym}, rtac @{thm predicate2I},
REPEAT_DETERM o
eresolve_tac [CollectE, exE, conjE, @{thm GrpE}, @{thm relcomppE}, @{thm conversepE}],
hyp_subst_tac ctxt,
rtac @{thm relcomppI}, rtac @{thm relcomppI}, rtac @{thm conversepI}, rtac @{thm GrpI},
rtac trans, rtac map_comp, rtac sym, rtac map_cong0,
REPEAT_DETERM_N n o rtac @{thm fst_fstOp},
in_tac @{thm fstOp_in},
rtac @{thm GrpI}, rtac trans, rtac map_comp, rtac map_cong0,
REPEAT_DETERM_N n o EVERY' [rtac trans, rtac o_apply,
rtac ballE, rtac subst,
rtac @{thm csquare_def}, rtac @{thm csquare_fstOp_sndOp}, atac, etac notE,
etac set_mp, atac],
in_tac @{thm fstOp_in},
rtac @{thm relcomppI}, rtac @{thm conversepI}, rtac @{thm GrpI},
rtac trans, rtac map_comp, rtac map_cong0,
REPEAT_DETERM_N n o rtac o_apply,
in_tac @{thm sndOp_in},
rtac @{thm GrpI}, rtac trans, rtac map_comp, rtac sym, rtac map_cong0,
REPEAT_DETERM_N n o rtac @{thm snd_sndOp},
in_tac @{thm sndOp_in},
rtac @{thm predicate2I},
REPEAT_DETERM o eresolve_tac [@{thm relcomppE}, @{thm conversepE}, @{thm GrpE}],
hyp_subst_tac ctxt,
rtac allE, rtac subst, rtac @{thm wppull_def}, rtac map_wppull,
CONJ_WRAP' (K (rtac @{thm wppull_fstOp_sndOp})) set_maps,
etac allE, etac impE, etac conjI, etac conjI, etac sym,
REPEAT_DETERM o eresolve_tac [bexE, conjE],
rtac @{thm relcomppI}, rtac @{thm conversepI},
EVERY' (map (fn thm => EVERY' [rtac @{thm GrpI}, rtac trans,
rtac trans, rtac map_cong0, REPEAT_DETERM_N n o rtac thm,
rtac (map_comp RS sym), atac, atac]) [@{thm fst_fstOp}, @{thm snd_sndOp}])] 1
end;
fun mk_rel_mono_strong_tac in_rel set_maps {context = ctxt, prems = _} =
if null set_maps then atac 1
else
unfold_tac [in_rel] ctxt THEN
REPEAT_DETERM (eresolve_tac [exE, CollectE, conjE] 1) THEN
hyp_subst_tac ctxt 1 THEN
unfold_tac set_maps ctxt THEN
EVERY' [rtac exI, rtac @{thm conjI[OF CollectI conjI[OF refl refl]]},
CONJ_WRAP' (K (etac @{thm Collect_split_mono_strong} THEN' atac)) set_maps] 1;
fun mk_map_transfer_tac rel_mono in_rel set_maps map_cong0 map_comp
{context = ctxt, prems = _} =
let
val n = length set_maps;
val in_tac = if n = 0 then rtac UNIV_I else
rtac CollectI THEN' CONJ_WRAP' (fn thm =>
etac (thm RS
@{thm ord_eq_le_trans[OF _ subset_trans[OF image_mono convol_image_vimage2p]]}))
set_maps;
in
REPEAT_DETERM_N n (HEADGOAL (rtac @{thm fun_relI})) THEN
unfold_thms_tac ctxt @{thms fun_rel_iff_leq_vimage2p} THEN
HEADGOAL (EVERY' [rtac @{thm order_trans}, rtac rel_mono, REPEAT_DETERM_N n o atac,
rtac @{thm predicate2I}, dtac (in_rel RS iffD1),
REPEAT_DETERM o eresolve_tac [exE, CollectE, conjE], hyp_subst_tac ctxt,
rtac @{thm vimage2pI}, rtac (in_rel RS iffD2), rtac exI, rtac conjI, in_tac,
rtac conjI,
EVERY' (map (fn convol =>
rtac (box_equals OF [map_cong0, map_comp RS sym, map_comp RS sym]) THEN'
REPEAT_DETERM_N n o rtac (convol RS fun_cong)) @{thms fst_convol snd_convol})])
end;
fun mk_in_bd_tac live surj_imp_ordLeq_inst map_comp map_id map_cong0 set_maps set_bds
bd_card_order bd_Card_order bd_Cinfinite bd_Cnotzero {context = ctxt, prems = _} =
if live = 0 then
rtac @{thm ordLeq_transitive[OF ordLeq_csum2[OF card_of_Card_order]
ordLeq_cexp2[OF ordLeq_refl[OF Card_order_ctwo] Card_order_csum]]} 1
else
let
val bd'_Cinfinite = bd_Cinfinite RS @{thm Cinfinite_csum1};
val inserts =
map (fn set_bd =>
iffD2 OF [@{thm card_of_ordLeq}, @{thm ordLeq_ordIso_trans} OF
[set_bd, bd_Card_order RS @{thm card_of_Field_ordIso} RS @{thm ordIso_symmetric}]])
set_bds;
in
EVERY' [rtac (Drule.rotate_prems 1 ctrans), rtac @{thm cprod_cinfinite_bound},
rtac (ctrans OF @{thms ordLeq_csum2 ordLeq_cexp2}), rtac @{thm card_of_Card_order},
rtac @{thm ordLeq_csum2}, rtac @{thm Card_order_ctwo}, rtac @{thm Card_order_csum},
rtac @{thm ordIso_ordLeq_trans}, rtac @{thm cexp_cong1},
if live = 1 then rtac @{thm ordIso_refl[OF Card_order_csum]}
else
REPEAT_DETERM_N (live - 2) o rtac @{thm ordIso_transitive[OF csum_cong2]} THEN'
REPEAT_DETERM_N (live - 1) o rtac @{thm csum_csum},
rtac bd_Card_order, rtac (@{thm cexp_mono2_Cnotzero} RS ctrans), rtac @{thm ordLeq_csum1},
rtac bd_Card_order, rtac @{thm Card_order_csum}, rtac bd_Cnotzero,
rtac @{thm csum_Cfinite_cexp_Cinfinite},
rtac (if live = 1 then @{thm card_of_Card_order} else @{thm Card_order_csum}),
CONJ_WRAP_GEN' (rtac @{thm Cfinite_csum}) (K (rtac @{thm Cfinite_cone})) set_maps,
rtac bd'_Cinfinite, rtac @{thm card_of_Card_order},
rtac @{thm Card_order_cexp}, rtac @{thm Cinfinite_cexp}, rtac @{thm ordLeq_csum2},
rtac @{thm Card_order_ctwo}, rtac bd'_Cinfinite,
rtac (Drule.rotate_prems 1 (@{thm cprod_mono2} RSN (2, ctrans))),
REPEAT_DETERM_N (live - 1) o
(rtac (bd_Cinfinite RS @{thm cprod_cexp_csum_cexp_Cinfinite} RSN (2, ctrans)) THEN'
rtac @{thm ordLeq_ordIso_trans[OF cprod_mono2 ordIso_symmetric[OF cprod_cexp]]}),
rtac @{thm ordLeq_refl[OF Card_order_cexp]}] 1 THEN
unfold_thms_tac ctxt [bd_card_order RS @{thm card_order_csum_cone_cexp_def}] THEN
unfold_thms_tac ctxt @{thms cprod_def Field_card_of} THEN
EVERY' [rtac (Drule.rotate_prems 1 ctrans), rtac surj_imp_ordLeq_inst, rtac subsetI,
Method.insert_tac inserts, REPEAT_DETERM o dtac meta_spec,
REPEAT_DETERM o eresolve_tac [exE, Tactic.make_elim conjunct1], etac CollectE,
if live = 1 then K all_tac
else REPEAT_DETERM_N (live - 2) o (etac conjE THEN' rotate_tac ~1) THEN' etac conjE,
rtac (Drule.rotate_prems 1 @{thm image_eqI}), rtac @{thm SigmaI}, rtac @{thm UNIV_I},
CONJ_WRAP_GEN' (rtac @{thm SigmaI})
(K (etac @{thm If_the_inv_into_in_Func} THEN' atac)) set_maps,
rtac sym,
rtac (Drule.rotate_prems 1
((box_equals OF [map_cong0 OF replicate live @{thm If_the_inv_into_f_f},
map_comp RS sym, map_id]) RSN (2, trans))),
REPEAT_DETERM_N (2 * live) o atac,
REPEAT_DETERM_N live o rtac (@{thm prod.cases} RS trans),
rtac refl,
rtac @{thm surj_imp_ordLeq}, rtac subsetI, rtac (Drule.rotate_prems 1 @{thm image_eqI}),
REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac CollectI,
CONJ_WRAP' (fn thm =>
rtac (thm RS ord_eq_le_trans) THEN' etac @{thm subset_trans[OF image_mono Un_upper1]})
set_maps,
rtac sym,
rtac (box_equals OF [map_cong0 OF replicate live @{thm fun_cong[OF sum_case_o_inj(1)]},
map_comp RS sym, map_id])] 1
end;
fun mk_trivial_wit_tac wit_defs set_maps {context = ctxt, prems = _} =
unfold_thms_tac ctxt wit_defs THEN HEADGOAL (EVERY' (map (fn thm =>
dtac (thm RS equalityD1 RS set_mp) THEN' etac imageE THEN' atac) set_maps)) THEN ALLGOALS atac;
end;